A invested Rs P+500 in compound interest for two years at 10% p.a. and received interest of Rs 840. If he invested Rs P at 15% p.a. for simple interest for the same period, then find the amount he received (in Rs).

The correct answer is Rs 4550.

We are given that A invested Rs (P + 500) at 10% p.a. compound interest for 2 years and earned an interest of Rs 840. We need to find the value of P first.

Step 1: Set up the Compound Interest equation

The formula for Compound Interest (CI) is:

$\mathrm{CI}=\mathrm{Principal}\times {(1+\frac{r}{100})}^n-\mathrm{Principal}$

Substituting the known values — Principal = (P + 500), r = 10%, n = 2 years, CI = 840:

$840=(P+500)\times {(1+\frac{10}{100})}^2-(P+500)$

$840=(P+500)\times {\left(\frac{11}{10}\right)}^2-(P+500)$

$840=(P+500)\times \frac{121}{100}-(P+500)$

Step 2: Factor out (P + 500)

$840=(P+500)\times (\frac{121}{100}-1)$

$840=(P+500)\times \frac{21}{100}$

Step 3: Solve for (P + 500)

$(P+500)=840\times \frac{100}{21}=4000$

So:

$P=4000-500=3500$

Step 4: Calculate Simple Interest on Rs P = 3500 at 15% p.a. for 2 years

The formula for Simple Interest is:

$\mathrm{SI}=\frac{P\times r\times t}{100}$

$\mathrm{SI}=\frac{3500\times 15\times 2}{100}=\frac{105000}{100}=1050$

Step 5: Calculate the total Amount received

$\mathrm{Amount}=\mathrm{Principal}+\mathrm{SI}=3500+1050=4550$

Therefore, the amount A received is Rs 4550.